THE MAN, NOT THE METHOD is algebraic; it is the root of 99900 x - 1 = 0 (as may be verified by the reader who remembers how to sum an infinite convergent geometrical progression). The first to prove that certain numbers are transcendental was Joseph Liouville (the same man who encouraged Hermite to write to Jacobi) who, in 1844, discovered a very extensive class of transcendental numbers, of which all those of the form where n is a real number greater than 1 (the example given above corresponds to n *= 10), are among the simplest. But it is probably a much more difficult problem to prove that a particular suspect, like e or K, is or is not transcendental than it is to invent a whole infinite class of transcendentals: the inventive mathematician dictates - to a certain extent - the working conditions, while the suspected number is entire master of the situation, and it is the mathematician in this case, not the suspect, who takes orders which he only dimly under- stands. So when Hermite proved in 1873 that e (denned a short way back) is transcendental, the mathematical world was not only delighted but astonished at the marvellous ingenuity of the proof. Since Hermite's time many numbers (and classes of numbers) have been proved transcendental. What is likely to remain a high-water mark on the shores of this dark sea for some time may be noted in passing. In 1934 the young Russian mathe- matician Alexis Gelf ond proved that all numbers of the type a6, where a is neither 0 nor 1 and b is any irrational algebraic number, are transcendental. This disposes of the seventh of David Hilbert's list of twenty-three outstanding mathematical problems which he called to the attention of mathematicians at the Paris International Congress in 1900. Note that ^irra- tionaP is necessary in the statement of Gelfond's theorem (if b — w/7?i, where n, m are rational integers, then a*, where a is any algebraic number, is a root of xm — an = 0, and it can be shown that this equation is equivalent to one in which all the coefficients are rational integers. 511